A Stochastic Segmentation Model for the Indentification of Histone Modification and DNase I Hypersensitive Sites in Chromatin

Abstract

Focal alterations in chromatin structure are essential for the proper functioning of various classes of transcriptional regulatory elements in the human genome. These changes can be detected through an increased sensitivity to DNase I and other nucleases due to an open and accessible chromatin conformation. Currently, quantitative analysis approaches use heuristic procedures to identify regions enriched for histone modifications and DNase I hypersensitivity. We here develop a stochastic segmentation model and associate inference framework to characterize the categorical and continuous features of hierarchical structures hidden in sequences. The proposed model has attractive statistical and computational properties and yields explicit formulas for posterior distribution of hidden states with a hierarchical structure. We propose an approximation method whose computational complexity is only linear in sequence length. We demonstrate the performance of the model via extensive simulations. We further use our model to identify DNase I sensitivity and DNase I hypersentitive sites over the Encyclopedia of DNA Elements (ENCODE) regions in human lymphoblastoid cells.

Appendices

Appendix A. Proof of 27.5 and 27.12

Proof of 27.5: To derive the mixture weight $\xi_{i,t}^{(k)}$, we first note that

$$f(\theta_t, y_t, s_{t-1}=k | {\cal F}_{t-1}) = \sum_{l=1}^K f(\theta_t, y_t, s_{t-1}=k, s_t=l | {\cal F}_{t-1}).\\ $$

When $l\neq k$,

$$\begin{aligned} &f(\theta_t, y_t, s_{t-1}=k, s_t=l | {\cal F}_{t-1})\\ =& f(\theta_t, y_t | {\cal F}_{t-1}, s_{t-1}=k, s_t =l)P(s_{t-1}=k, s_t =l | {\cal F}_{t-1}) \\ =& f(y_t | {\cal F}_{t-1}, J^{(l)}_{t}=t)f(\theta_t| {\cal F}_{t}, J^{(l)}_{t}=t)P(s_t =l| s_{t-1}=k)P(s_{t-1}=k| {\cal F}_{t-1}) \\ =& f(y_t | {\cal F}_{t-1}, J^{(l)}_{t}=t)f(\theta_t| {\cal F}_{t}, J^{(l)}_{t}=t)p_{k,l} \xi_{t-1}^{(k)}.\\ \end{aligned}$$

When $l= k$,

$$\begin{aligned} &f(\theta_t, y_t, s_{t-1}=k, s_t=k | {\cal F}_{t-1})= \sum_{i=1}^{t-1} f(J^{(k)}_{t}=i, \theta_t, y_t|{\cal F}_{t-1})\\ =& \sum_{i=1}^{t-1} f(\theta_t, y_t|{\cal F}_{t-1},J^{(k)}_{t}=i)P(s_{t-1}=k, s_t =k | {\cal F}_{t-1}) \\ =& \sum_{i=1}^{t-1} f(y_t | {\cal F}_{t-1},J^{(k)}_{t}=i)f(\theta_t| {\cal F}_{t}, J^{(k)}_{t}=i)P(s_t =k| s_{t-1}=k)P(s_{t-1}=k| {\cal F}_{t-1}) \\ =& \sum_{i=1}^{t-1} f(y_t | {\cal F}_{t-1}, J^{(l)}_{t}=t)f(\theta_t| {\cal F}_{t}, J^{(k)}_{t}=i)p_{k,k} \xi_{i, t-1}^{(k)}.\\ \end{aligned}$$

Let

$$\xi_{i,t}^{(k)*} = \left\{\begin{array}{ll} \big( \sum_{l \neq k} \xi_{t-1}^{(l)} p_{lk} \big) f(y_t | J_t^{(k)}=t) & i=t, \\ p_{kk} \xi_{i, t-1}^{(k)} f(y_t | {\cal F}_{t-1}, J_t^{(k)}=i) & i < t.\end{array} \right.$$

We then have

$$f({\boldmath \beta}_t | {\cal F}_t) \propto \sum_{k=1}^K \xi_{t,t}^{(k)*} f(\theta_t| {\cal F}_{t}, J^{(l)}_{t}=t) + \sum_{k=1}^K \sum_{i=1}^{t-1} \xi_{i,t}^{(k)*}f(\theta_t| {\cal F}_{t}, J^{(k)}_{t}=i).$$

Hence, the mixture weight $\xi_{i,t}^{(k)}$ is the conditional probability which can be determined via normalization of $\xi_{i,t}^{(k*)}$. Furthermore, simple algebra shows that

$$f(y_t | J_t^{(k)}=t)=\psi_{0,0}^{(k)} \big/ \psi_{t,t}^{(k)}, \qquad f(y_t | {\cal F}_{t-1}, J_t^{(k)}=i)=\psi_{i,t-1}^{(k)} \big/ \psi_{i,t}^{(k)},$$

where

$$\psi_{0,0}^{(k)}=(\kappa^{(k)})^{-\frac{1}{2}} \frac{(\lambda^{(k)})^{-g^{(k)}}}{\Gamma(g^{(k)})}, \qquad \psi_{ij}^{(k)}=(\kappa_{ij}^{(k)})^{-\frac{1}{2}} \frac{(\lambda_{ij}^{k})^{-g_{ij}^{(k)}}}{\Gamma(g_{ij}^{(k)})},$$

for $i\le j$. This proves (27.5).

Proof of (27.12): We use Bayes’ theorem to combine the forward filter (27.4) with its backward variant (27.10) to derive the posterior distribution of θ _t given ${\cal F}_T$ $(1\le t < T)$

$$\begin{aligned}f(\theta_t|{\cal F}_T) = \sum_{k=1}^K f(\theta_t, s_t=k|{\cal F}_T) \propto \sum_{k=1}^K f(\theta_t, s_t=k|{\cal F}_t)\frac{ f(\theta_t, s_t=k| {\cal F}_{t+1,T})}{f(\theta, s_t=k)}.\end{aligned}$$

(27.22)

We first consider the following:

$$\begin{aligned}&f(\theta_t, s_t=k|{\cal F}_t) f(\theta_t, s_t=k|{\cal F}_{t+1,T}) \big/ f(\theta, s_t=k) \notag\\ =&\frac{\sum_{i=1}^t\xi_{i,t}^{(k)}f(\theta_t | {\cal F}_{i,t})\cdot\{\widetilde{q}_{kk}\sum_{j=t+1}^T\eta_{t+1,j}^{(k)}f(\theta_t |{\cal F}_{t+1,j})+\sum_{l\neq k}\widetilde{q}_{lk}\eta_{t+1}^{(l)}f(\theta_t|s_t=k)\}}{P(s_t=k)f(\theta_t|s_t=k)} \notag\\ =&\frac{\sum_{i=1}^t\xi_{i,t}^{(k)}f(\theta_t | {\cal F}_{i,t})\cdot \widetilde{q}_{kk}\sum_{j=t+1}^T\eta_{t+1,j}^{(k)}f(\theta_t |{\cal F}_{t+1,j})}{\pi_k f(\theta_t|s_t=k)}\notag\\ & \hspace{20pt}+\frac{\sum_{i=1}^t\xi_{i,t}^{(k)}f(\theta_t | {\cal F}_{i,t})\cdot \sum_{l\neq k}\widetilde{q}_{lk}\eta_{t+1}^{(l)}f(\theta_t|s_t=k)}{\pi_k f(\theta_t|s_t=k)}\notag\\ =&\sum_i^t\xi_{i,t}^{(k)}\sum_{l\neq k}\frac{\widetilde{q}_{lk}}{\pi_k}\eta_{t+1}^{(l)}f(\theta_t | {\cal F}_{i,t})+\frac{\widetilde{q}_{kk}}{\pi_k}\sum_{1 \le i \le t \le j \le T}\xi_{i,t}^{(k)}\eta_{t+1,j}^{(k)}\frac{f(\theta_t | {\cal F}_{i,t})f(\theta_t | {\cal F}_{t+1,j})}{f(\theta_t|s_t=k)}.\end{aligned}$$

Note that

$$\frac{f(\theta_t | {\cal F}_{i,t})f(\theta_t | {\cal F}_{t+1,j})}{f(\theta_t|s_t=k)}=\frac{\psi_{i,t}^{(k)}\psi_{t+1,j}^{(k)}}{\psi_{i,j}^{(k)} \psi_{0,0}^{(k)}}f(\theta_t | {\cal F}_{i,j}),$$

Hence, (27.12) is proved.

Appendix B. EM Algorithm for Hyperparameter Estimation

The inference procedure in the above sections involve the hyperparameters $\Phi=$ $\{ Q, z^{(k)}$, $\kappa^{(k)}$, $\lambda^{(k)}$, $g^{(k)}$; $k=1, \dots, \}$, is a $[4K+K(K-1)]$-dimensional vector. We can use the EM algorithm to exploit the much simpler structure of the log likelihood $l_c(\Phi)$ of the complete data $\{(y_t, s_t, \theta_t), 1\le t \le T\}$, which is expressed as

$$\begin{aligned}& l_c(\Phi) = \sum_{t=1}^T \log f(\{y_t, s_t, \theta_t \}) \nonumber\\ = &\sum_{t=1}^T \Big\{ \log f(y_t|\theta_t) + \sum_{k=1}^K f(\theta_t | s_t=k) {\bf 1}_{ \{ s_t = k \}} + \sum_{k,l=1}^K \log (p_{kl}) {\bf 1}_{ \{ s_{t-1} = k, s_t = l \} } \Big\}\nonumber\\ = &-\sum_{t=1}^T \Big\{ \frac{(y_t-\mu_t)^2}{2\sigma_t^2}+\frac{1}{2}\log(2\sigma_t^2)\Big\} -\sum_{t=1}^T\sum_{k=1}^K\Big\{\frac{(\mu_t-z^{(k)})^2}{ 2\sigma_t^2 \kappa^{(k)}}+ \frac{1}{2}\log( 2\sigma_t^2 \kappa^{(k)})\Big\}\nonumber\\ &-\sum_{t=1}^T\sum_{k=1}^K\Big\{g^{(k)}\log(\lambda^{(k)})- \log(\Gamma(g^{(k)}))-(g^{(k)}-1)\log (2\sigma_t^2)+ \frac{1}{2\sigma_t^2\lambda^{(k)}}\Big\} {\bf 1}_{\{s_t=k\}}\nonumber\\ &+ \sum_{t=1}^T \sum_{k,l=1}^K \log (p_{kl}) {\bf 1}_{ \{ s_{t-1} = k, s_t = l \} }.\end{aligned}$$

(27.23)

The E-step of the EM algorithm calculates $E[l_c(\Phi)|{\cal F}_t]$, which involves the computation of the conditional expectations:

$$E\Big[\frac{ (y_t-\mu_t)^2}{2\sigma_2^2} |{\cal F}_T\Big],\qquad E[\log(2\sigma_t^2)|{\cal F}_T],\qquad E\Big(\frac{(\mu_t-z^{(k)})^2}{2\sigma_t^2\kappa^{(k)}} {\bf 1}_{\{s_t=k\}}|{\cal F}_T\Big),$$

$$E[\log( 2\sigma_t^2\kappa^{(k)}) {\bf 1}_{\{s_t=k\}}|{\cal F}_T], E[\log (2\sigma_t^2) {\bf 1}_{\{s_t=k\}}|{\cal F}_T], E\Big(\frac{1}{2 \sigma_t^2 \lambda^{(k)}} {\bf 1}_{\{s_t=k\}}|{\cal F}_T\Big),$$

and the conditional probability:

$$P(s_t = k|{\cal F}_T), \qquad P(s_{t-1} = k, s_t = l|{\cal F}_T).$$

The M-step of the EM algorithm involves calculating the partial derivatives of $E[l_c(\Phi)|{\cal F}_t]$ with respect to Φ. Simple algebra yields the following updating formulas for Φ:

$$\begin{aligned}\widehat{q}_{kl, \rm new} =& \frac{\sum_{t=2}^T P(s_{t-1}=k, s_t=l|{\cal F}_T, \widehat{\Phi}_{\rm old})}{\sum_{t=2}^T P(s_{t-1}=k|{\cal F}_T, \widehat{\Phi}_{\rm old})},\notag\end{aligned}$$

$$\begin{aligned} \widehat{z}^{(k)}_{\rm new} =& \frac{\sum_{t=1}^T E[\mu_t/(2\sigma_t^2) {\bf 1}_{ \{ s_t = k \}} |{\cal F}_T, \widehat{\Phi}_{\rm old}]}{\sum_{t=1}^T E[P_t {\bf 1}_{\{s_{t}=k\}}|{\cal F}_T, \widehat{\Phi}_{\rm old}]},\notag \end{aligned}$$

$$\begin{aligned} \widehat{\kappa}^{(k)}_{\rm new} =& \frac{2\sum_{t=1}^T E[(\mu_t - \widehat{z}^{(k)}_{\rm old})^2/ (2\sigma_t^2) {\bf 1}_{ \{ s_t = k \}} |{\cal F}_T, \widehat{\Phi}_{\rm old}]}{\sum_{t=1}^T E[{\bf 1}_{\{s_{t}=k\}}|{\cal F}_T, \widehat{\Phi}_{\rm old}]}, \notag\end{aligned}$$

$$\begin{aligned} \widehat{\lambda}^{(k)}_{\rm new} =& \frac{\sum_{t=1}^T E[(2\sigma_t)^{-1} {\bf 1}_{\{s_t=k\}}|{\cal F}_T, \widehat{\Phi}_{\rm old}]}{ \sum_{t=1}^T g^{(k)}_{\rm old}E[ {\bf 1}_{\{s_t=k\}} |{\cal F}_T, \widehat{\Phi}_{\rm old}].}\end{aligned}$$

(27.24)

I-terms in (27.24) can be obtained as follows:

$$E\Big( \frac{\mu_t}{2\sigma_t^2} {\bf 1}_{ \{ s_t = k \}} \Big|{\cal F}_T, \widehat{\Phi}_{\rm old}\Big) = \sum_{t=1}^T \alpha^{(k)}_{ijt} E \Big( \frac{\mu_t}{2\sigma_t^2} |C_{ij}^{(k)}, {\cal F}_T, \widehat{\Phi}_{\rm old}\Big),$$

(27.25)

$$E\Big( \frac{\mu_t}{2\sigma_t^2} \Big|C_{ij}^{(k)}{\cal F}_T, \widehat{\Phi}_{\rm old} \Big) = \lambda_{ij}^{(k)} g_{ij}^{(k)} z_{ij}^{(k)},$$

(27.26)

$$E\Big( \frac{1}{2\sigma_t^2} {\bf 1}_{\{s_{t}=k\}} \Big|{\cal F}_T, \widehat{\Phi}_{\rm old} \Big) = \sum_{1 \le i \le t \le j \le T} \alpha_{ijt}^{(k)} g_{ijt}^{(k)} \lambda_{ijt}^{(k)}.$$

(27.27)

$$\begin{aligned} E\Bigg( &\frac{\mu_t-z_{\rm old}^{(k)})^2}{2\sigma_t^2}{\bf 1}_{\{s_t=k\}} \Big|{\cal F}_T, \widehat{\Phi}_{\rm old}\Bigg) = \sum\limits_{1 \leq i \leq t \leq j \leq T}\alpha_{ijt}^{(k)} \Bigg\{E \left(\frac{\mu_t^2}{2\sigma_t^2}\Big|C_{ij}^{(k)}, {\cal F}_T,\widehat{\Phi}_{\rm old} \right) \nonumber\\&- 2 z_{\rm old}^{(k)}E \left( \frac{\mu_t}{2\sigma_t^2}\Big|C_{ij}^{(k)}, {\cal F}_T, \widehat{\Phi}_{\rm old} \right)+(z_{\rm old}^{(k)})^2) E\left( \frac{1}{2\sigma_t^2}\Big|C_{ij}^{(k)}, {\cal F}_T, \widehat{\Phi}_{\rm old}\right)\Bigg\}, \end{aligned}$$

(27.28)

in which

$$\begin{aligned} E\left( \frac{\mu_t^2}{2\sigma_t^2} \Big|C_{ij}^{(k)}, {\cal F}_T,\widehat{\Phi}_{\rm old}\right)=\frac{\kappa_{ij}^{(k)}}{2}+\lambda_{ij}^{(k)}g_{ij}^{(k)}\left(z_{ij}^{(k)}\right)^2.\end{aligned}$$

(27.29)

We can use the BCMIX approximations instead of the full recursions to determine the items (27.25)–(27.29) in order to speed up computation. The iteration scheme (27.24) is carried out until convergence to estimate hyperparameters.